Relaxation and time scales in disordered XX model with on-site dephasing

The relaxation of observables to their non-equilibrium steady states in a disordered XX chain subjected to dephasing at every site has been intensely studied in recent years. In our work, we seek to demystify the relaxation of `Imbalance' in such a system, using a comprehensive analysis supported by analytics, and numerical simulations at large system sizes. Using two/three-site models, we can analytically predict emergence of several timescales in the system and extract results which match with numerics very closely without any extra fitting parameters. Then for long timescales, studying the low-energy eigenspectrum of the Liouvillian shows us that the decay is a power law, akin to clean systems, with the exponent depending on factors such as boundary conditions, which matches with exact numerical data. The final phase of relaxation is governed by the relevant lowest magnitude Liouvillian gap, as expected for finite sized systems.